<h2>Problem 232</h2>
<div style="color:#666;font-size:80%;">13 February 2009</div><br />
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<p>Two players share an unbiased coin and take it in turns to play &quot;The Race&quot;. On Player 1's turn, he tosses the coin once: if it comes up Heads, he scores one point; if it comes up Tails, he scores nothing. On Player 2's turn, she chooses a positive integer <var>T</var> and tosses the coin <var>T</var> times: if it comes up all Heads, she scores 2<img src="" style="display:none;" alt="^(" /><sup><var>T</var>-1</sup><img src="" style="display:none;" alt=")" /> points; otherwise, she scores nothing. Player 1 goes first. The winner is the first to 100 or more points.</p>

<p>On each turn Player 2 selects the number, <var>T</var>, of coin tosses that maximises the probability of her winning.</p>

<p>What is the probability that Player 2 wins?</p>

<p>Give your answer rounded to eight decimal places in the form 0.<i>abcdefgh</i> .</p>
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